Create invisibility cloak (alpha mask material)
After reading Leon Cheung's answer, I've redone my solution:

Say you have this setup



and you want to make the plane into an invisibility cloak that affects only the red monkey.

First, make the background transparent (see here for help). 

For Blender Render, select the plane, and under the Materials tab of Properties window, enable Transparency, select Mask and set Alpha to zero:



For Cycles Render, select the plane and set its material to Holdout,



For both Blender and Cycles Render, you create two render layers: one with the invisibility cloak and objects affected by it, and one with the normal, unaffected objects.

Then create this node setup:



and the result is: